Question

PLEASE HELP OMG calculate the slope of the graph below at the point (-1,5)

f(x) = x^5 + 1/x^3 +7

Answer 1

The slope of the graph at x=-1 is 2

Explanation:

Instant rate of change

Given a real function f(x), the instant rate of change with respect to x is defined as the derivative of f which coincides with the instant value of the slope of the tangent line in the point (x,y).

We have the function:

`\displaystyle f(x) = x^5+(1)/(x^3)+7`

Prepare the function to apply the power rule of the derivative:

`\displaystyle f(x) = x^5+x^(-3)+7`

Recall the power rule:

`(x^n)' = nx^(n-1)`

Also, the derivative of a constant is zero.

Taking the derivative:

`\displaystyle f'(x) = 5x^4-3x^(-4)`

Evaluating for x=-1:

`\displaystyle f'(-1) = 5(-1)^4-3(-1)^(-4)`

`\displaystyle f'(-1) = 5*1-3*1=2`

The slope of the graph at x=-1 is 2